I don't recall coming across the word myself before, but starting from...
- happening as part of an official ceremony or celebration when someone (such as a newly elected official) begins an important job.
- happening as part of an inauguration.
- happening as the first one in a series of similar events (emphasis mine)
...I figured I was bound to find...
occurring at the close of a term of office — opposed to inaugural
I think an exaugural president, for example, would naturally be understood to mean a president leaving office in contexts where a new president is due to take over. But in my understanding, whereas that new person would be an inaugural president, an inaugurating president would more naturally be understood to mean the first ever president [of a newly-created office].
So even though I can't find a specific dictionary definition to back me up on this one, I suggest...
exaugurating2 [president, board, etc.]
1 Source: Merriam-Webster
2 Not in MW. OED has the following obsolete/rare entry, but I think since so few people would be aware of the word, it wouldn't jar with my slightly creative (but obvious in context) redefinition...
To cancel the inauguration of; to unhallow, make profane.
To augur evil to.